The Correct Answer and Explanation is:
The correct answer is 5/3.
The number 1.6666666667 is the decimal representation of a repeating decimal, which has been rounded for display. The actual number is 1.666…, where the digit 6 repeats infinitely. To convert this repeating decimal into a fraction, we can use an algebraic method.
First, let the variable x equal the repeating decimal:
x = 1.666…
The goal is to eliminate the repeating part of the decimal. Since only one digit is repeating (the 6), we can multiply the equation by 10. This shifts the decimal point one place to the right:
10x = 16.666…
Now we have two equations. If we subtract the first equation from the second, the repeating decimal portion will cancel out:
10x = 16.666…
– x = 1.666…
9x = 15
Next, we solve for x by dividing both sides by 9:
x = 15/9
This gives us a fraction, but the problem asks for the fraction in its simplest form. To simplify 15/9, we need to find the greatest common divisor (GCD) of the numerator (15) and the denominator (9). The factors of 15 are 1, 3, 5, and 15. The factors of 9 are 1, 3, and 9. The greatest common divisor is 3.
We can simplify the fraction by dividing both the numerator and the denominator by their GCD, which is 3:
15 ÷ 3 = 5
9 ÷ 3 = 3
Therefore, the fraction in its simplest form is 5/3.
Alternatively, you could recognize that 1.666… is the same as 1 + 0.666… The repeating decimal 0.666… is a common fraction, 2/3. So, you can calculate the result as 1 + 2/3. To add these, you convert 1 into 3/3, which gives you 3/3 + 2/3 = 5/3.
